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W 1,1(Ω) Solutions of Nonlinear Problems with Nonhomogeneous Neumann Boundary Conditions

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Abstract

In this paper we study the existence of W 1,1(Ω) distributional solutions of the nonlinear problems with Neumann boundary condition. The simplest model is

$$\left \{ \begin{array}{cc} -\Delta_{p}u + |u|^{s-1}u = 0, & {\rm in}\, \Omega;\\ |\nabla u|^{p-2}\nabla u . \eta = \psi, & {\rm on} \, \partial\Omega;\end{array}\right.$$

where Ω is a bounded domain in \({I\!R^{N}}\) with smooth boundary \({\partial\Omega, 1 < p < N, s > 0, \eta}\) is the unit outward normal on \({\partial\Omega {\rm and} \psi \in L^{m}(\partial\Omega), m > 1}\).

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Authors and Affiliations

  1. Dipartimento di Matematica, “Sapienza” Università di Roma, P.le A. Moro 5, 00185, Roma, Italy

    Lucio Boccardo

  2. Departamento de Análisis Matematico, Universidad de Granada, Av. Fuentenueva S/N, 18071, Granada, Spain

    Lourdes Moreno-Mérida

Authors
  1. Lucio Boccardo
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  2. Lourdes Moreno-Mérida
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Correspondence to Lourdes Moreno-Mérida.

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Dedicated to our friend “Mazon” for his 70th birthday: y sólo entonces había comprendido que un hombre sabe cuando empieza a envejecer porque empieza a parecerse a su padre ([33])

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Boccardo, L., Moreno-Mérida, L. W 1,1(Ω) Solutions of Nonlinear Problems with Nonhomogeneous Neumann Boundary Conditions. Milan J. Math. 83, 279–293 (2015). https://doi.org/10.1007/s00032-015-0235-0

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